Alexander’s Walk Home
This problem is from Alexander Karabegov.
Suppose that when Alexander walks home he always chooses one or the other of two routes. In one route he stays on a sidewalk that makes a right-angle turn and is tangent to a circular flower garden on his right in two places (see the figure below). He chooses an alternate route when he wants to smell the flowers on the other side of the flower garden. In the alternate route, he leaves the sidewalk and walks in a straight line that is tangent to the garden, which is now on his left. After smelling the flowers he continues in the same straight line he started walking (off the sidewalk) until he reaches the sidewalk that is around the corner.
There are two things about Alexander:
- He does not always exit the sidewalk in the same place when walking with the flower garden on his left, and
- He wears a pedometer that tells him exactly how many steps he takes while walking home.
Being an observant person, he notices that the number of steps he takes when walking with the flower garden on his left is always 40 steps less than when he stays on the sidewalk, regardless of where he exits the sidewalk. What is the diameter of the circular flower garden, assuming that Alexander’s steps are always 1 yard long?
Please submit all work to mathpotw@acu.edu before Thursday, September 19 @ 5:00 PM.
Solution to Alexander’s Walk Home
Correct solutions were submitted by: Wyatt Witemeyer and Bethany Witemeyer
By referring to the figure below, we can see that the segment AC that Alexander walks across the grass can be split into two pieces, a segment AE plus a segment EC. Now the segment AE is equal in length to the segment AD, and the segment EC is equal in length to the segment FC. So the extra amount that Alexander walks when he stays on the sidewalk is just the “corner” which has length twice the radius of the circle.
Therefore the diameter of the circle is 40 yards.
